Piecewise Function: Solving Options Assignment

[vladimir69]

hi,
i have been working on an Options assignment, and i have this formula as one of my answers:

$$X_{t}=\left\{\begin{array}{cc}X_{t-1},&\mbox{if } S_{t} < S_{t-1}\\aS_{t},& \mbox{if } S_{t}\geq S_{t-1}\end{array}\right$$

where a is a constant. the question however asks for a "single equation", would the equation(s)? i have defined above be ok? is there a way to rewrite this not using a piecewise function. hopefully you know what i mean.

thnx
vladimir

 

[honestrosewater]

Just a thought- find the definition of "equation" you were given. If you weren't given one...
As for another way to write a piecewise function, why not define a function on each of your intervals (the restriction of the piecewise function to each interval) and define the piecewise function as the union of those functions? I.e., given abs: R --> R,
$$\mbox{abs}(x) = \left\{\begin{array}{cc}x,&\mbox{if } x \geq 0 \\-x,& \mbox{if } x < 0 \end{array}\right$$
Let f: R⁺ --> R, f(x) = x and g: R^- --> R, g(x) = -x. Then abs = f U g - which is almost certainly a single equation ;) Doesn't save time, but I think it works.

 

[tacman]

Hi Vladimir,

Great job on your Options assignment! Your formula looks correct and it is a valid piecewise function. However, if the question specifically asks for a "single equation", then you may need to rewrite it without using a piecewise function. Here are a few options you can explore:

1. Use absolute value notation: Instead of using the "if" condition, you can use absolute value notation to represent the two cases. Your equation would then be:

X_t = |aS_t| + |(1-a)S_{t-1}|

2. Use a conditional statement: You can also rewrite the equation using a conditional statement, such as "if-else" or "if-then-else". Your equation would then be:

X_t = \begin{cases} X_{t-1}, & \mbox{if } S_t < S_{t-1}\\ aS_t, & \mbox{if } S_t \geq S_{t-1} \end{cases}

3. Use a piecewise function with different cases: Another option is to rewrite the equation using a piecewise function with different cases, so that each case only has one condition. Your equation would then be:

X_t = \begin{cases} X_{t-1}, & \mbox{if } S_t < S_{t-1}\\ aS_t, & \mbox{if } S_t = S_{t-1}\\ aS_t + (1-a)S_{t-1}, & \mbox{if } S_t > S_{t-1} \end{cases}

Ultimately, the best way to rewrite the equation without using a piecewise function would depend on the specific requirements and context of your assignment. I hope these suggestions help and good luck with your assignment!